JOURNAL
OF FUNCTIONAL
77, 434460
ANALYSIS
(1988)
The Nonaccretive
Radiative Transfer Equations:
Existence of Solutions and Rosseland Approximation
C. BARDOS, *. ’ F. GOLSE,* B. PERTHAME,*
* hole
AND R. SENTIS*
Normale
SupPrieure,
Centre de MathPmatiques
Appliquies,
4.5, rue d’iilm,
75230 Paris Ckdex OS,
i UniversitP Paris XIII, DPpartement
de MathPmatiques,
Au. J. B. Ckment,
93430 Villetaneuse,
and
: C. E. A. Lime&Valenton,
DPpartement
MA, See MCN,
B. P. 27, 94190 Villeneuve-Saint
Georges, France
Communicated
Received
We present an existence
transfer equations
theory
by H. B&is
December
and
22. 1986
an asymptotic
au, n.v,u,
u(z-i,)
?-I+ -+-+u,-i+o
E
21,I,ixx.~,
=k
analysis
for
the radiative
in X,
(1)
u,l,=“=u”,
where
u,, = u,(t, x, Q),
I E iw,,
x E X t Iw”’ ‘, .Q E S’v, and
fi,,(l, .x) =
l/l SN) s u,(t, x, Q) dQ. We prove that, even if CJ has a singularity
(u(0) = + OI), (1)
hasasolutionu,~L”(R+xX~S~).Ass
+ 0, we show that U, converges pointwise
to a function
u E L’( R + x X), solution of the degenerate
parabolic
equation
~-dF(+O
This is achieved without
any monotonicity
cannot use the theory of nonlinear
contraction
in X,
assumption
semigroups.
on 0 and
therefore
‘$3 1988 Academic
one
Press. Inc.
We consider the problem of the existence of solutions and the diffusion
approximation
for the Radiative
Transfer (R.T.) equations. These
equations describe the transport of photons in a starlike medium and are,
mathematically,
a nonlinear version of the transport of neutrons. The
results that we prove here are wellknown in the linear case: the existence
follows from the study of the spectrum of the transport operator, which has
been carried out by many authors [l, 3, 8, 12, 19, 26, 281, and the dif434
0022-1236/88
$3.00
Copyright CC 1988 by Academic Press, Inc.
All rights of reproduction
in any form reserved
RADIATIVE
TRANSFER
EQUATIONS
435
fusion approximation has also been studied extensively [ 3, 7, 12, 20, 21, 26,
271. The R.T. equation has been studied more recently and these results
(existence of solutions and Rosseland approximation)
are known only
under some monotonicity
assumptions on the opacity (the term r~ in the
equation below), which ensure that the R.T. operator is accretive [2, 17,
221. Our purpose here is to prove these results without any accretiveness
assumptions since they have no physical meaning (see [lo, 14, 23, 251).
This is achieved for the “grey problem,” which reads, for a fixed number
E> 0.
(1)
or, in a stationary case,
(2)
where U, 5 ~,(t, X, 0) (resp. U, c u,(x, Q)) with x E X (some smooth bounded convex open subset of RN+‘), R E SN (the unit sphere of RN+‘), t 2 0.
In (l), (2), v”denotes the integral
where df2 is the normalized
Lebesgue measure on SN (jS~ dQ = 1) and
(axxsN)~=((x,a)E~XXSN,~(X)~~<o},
where n(x) is the outward normal to X at the point x of 8.X Finally, let us
point out that the positive function C(T) may have a singularity at T= 0:
o(O) = + CE which is one of the main difficulties of the model.
It is wellknown, from physical arguments, that, as E goes to 0, U, converges to a function u(x) which is the solution of a nonlinear degenerate
parabolic (resp. elliptic) equation
(3)
436
BARDOS
ET AL.
or
iv - AF(v) = 0,
v I ax = k
(4)
at least when k is constant and uO= uO(x). This is called the “Rosseland
approximation.”
The function F is given by
1
1
F(T)=----.
NS 1 a(T)
We will study here Eqs. (l), (2) and their approximations
(3), (4). First
let us notice that when k, uO,f are nonnegative, any solution U, is also nonnegative and that the maximum principle holds so that U, is bounded a
priori. However, this is not enough to get the existence of solutions of (1)
or (2) because of the nonlinear term and because the expression
a(ii,)(u, - ii,) has no clear meaning when ii, = 0. Thus we will need a
stronger regularity property. It is given by a compactness result introduced
in [ 16, IS] that we will use in an essential way, first to prove the existence
of solutions of (1 ), (2) by Schauder’s fixed point theorem and then to get
the strong convergence of u,: as E goes to 0.
This paper is organized as follows. In Section I, we study the existence of
solutions of (1 ), (2). In Section 11, we prove the Rosseland approximation.
In Section III, we study the existence of solutions for a more complete
model than (1 ), (2) (model with frequency),
nu+Q.v.,u+a,.(T)(u-B,(T))=g,
IIT+
s08‘
a,,( T)( B,( T) - ii) dv =f,
where the unknown functions are now T(x) and u(x, 8, v), where VE iw,
denotes the frequency of the photons. The treatment of this system uses a
generalization of the methods developed in Sections I and II. However, it
requires more technical arguments and, for the sake of simplicity, we give
only partial results (we do not study the evolution equation or the case
/z= 0). Finally, let us recall that the Rosseland approximation
for the
model with frequency is still an open question when 0 and B do not satisfy
monotonicity assumptions as in [2].
437
RADIATIVE TRANSFER EQUATIONS
I. EXISTENCE OF SOLUTIONS
1.1. Assumptions and Main Results
Before stating our main existence results, let us say more precisely what
is a solution of (1) or (2) and introduce some assumptions. First, the
function (TE C( R + *, 03+ *) is supposed to satisfy
3P>l,
3c,
3~,,
0 < om6 a(u) < c/u”p,
vu d 1,
(AlI
and (to treat the case il = 0)
3u,, such that for u > u,, Q is nonincreasing,
decreasing, and O(U) -+U+w 0.
CJ(U)u is non(AZ)
In particular, from (Al ) we deduce that U(U) u -+ 0 as u -+ 0, and thus the
term a( ii) u always has meaning if ii E L”(X), u > 0, since we may take as a
convention a(O) 0 = 0.
Finally, since we treat the existence of solutions of (l), (2), we choose
E= 1 in this section to simplify notations.
THEOREM
1 (Existence
for
the
Stationary
assumption (Al), let f 20, k 2 0 belong to L”(Xx
1= 0 and (A2) holds, the equation
Problem).
Under
the
SN); then, if ,I > 0, or if
Au+ f2.v,u + a(fi)(u- ii)=f,
4(?X..s~,=k
(5)
has at least one nonnegative solution u E L”(Xx SN) such that Q .V,u E
Lp(Xx SN), a(G) UE LP(Xx SN), and a(G) ii~ L”(X).
THEOREM 2 (Existence
for the Evolution
Problem).
Under the
assumption (A 1), let u0 3 0, k >, 0 belong to Lz(dX x SN), then the equation
~+Q.v~~u+o(ri)(u-ir)=o,
= k,
4t,WxsN)
(6)
uI,=o=ug
has at least one generalized nonnegative bounded solution UE C( [0, T];
LP(Xx s”))nL”(R+
xxx SN). Moreover au/at + Q . V,u and a(c) u
belong to L”([W+; LP(Xx SN)).
Remarks. (1) With the assumption (Al), (1) is not a Lipschitz perturbation of the transport operator in L’; this motivates Theorem 2.
438
BARDOSET AL.
(2) The monotonicity assumption which implies the accretiveness of
the R.T. operator is that g is nonincreasing and c(T) T is nondecreasing.
This set of assumptions
has been introduced
by Mercier
[22].
Assumption (A2) is weaker.
(3) In (5) or (6) the boundary condition holds in the sense of traces
for the transport space. See [9] for weaker assumptions on k.
Before proving these theorems, let us recall some compactness results
which are the cornerstone of the proof.
PROPOSITION 1 [16].
(1) Let UEL~(XXS”‘),
q>l,
and Q.V,UE
LY(Xx S”), 241
c,~xxs~,~=k~Lm((aXxSN)_).
Then GE V”(X)
for any s,
0 < s < inf( l/q, 1 - l/q), and
(2) Let UEL~([O,T]XXXS~),
q>l,
and du/LYt+D.V,uE
Ly([O, T]xXxSN),
and ul,=,=u,~L”(XxS~),
and uIcRXxS1vjm=k~
L”([O, T]x(~YXXS”)~).
Then ii~l%‘~~~([O.T]xX)
for any s, O<s<
inf( l/q, 1 - l/q), and
IIii IIwl.yd ck ~d( IIu IILq+ IIau/at+ D .V,U IId.
Here we will use only the easy consequence that in both cases il is compact
in the corresponding Ly spaces.
2. Proof of Theorems 1 and 2.
We prove Theorem 1 in three steps. First we study the case where I”>0
of 0 and we
and 0 remains bounded (i.e., we make a regularization
consider o,,,(T) = CT(T+ l/M) in place of a), then we let i tend to 0, and
finally we let M tend to infinity.
(i) Case of 1”> 0, CJBounded. Here we study the existence of solutions
of (5) with 0 replaced by o,+,(T) = a( T+ l/M) with M> 0. Thus for
TEL”(X),
let us introduce the equation
~u+52.V,u+o,(T)(u-~)=f,
0I(axx.P- = k.
(7)
It is well known that this linear transport equation has a unique solution
UEL~(XXS”‘),
such that Q .V,VE L”(XxSN).
Moreover, since 2 is
positive we have
OdudMax
(8)
RADIATIVE
TRANSFER
439
EQUATIONS
Let us define the set
,
and V, the operator defined by ‘$(T) = u’ for TE S, and v being the solution
of (7). %?is clearly continuous for the L’ topology from the convex set S
into itself. Moreover, since we have for any v = g(T), TE S,
IQ.Vxul
G
Ilf
IIL~+(~+2w
(
IIf IILX
lML=+~
>
3
we may apply Lemma 1 and obtain that the range of %?is bounded in
H”‘(X) and thus is compact in L2(X). This shows, using Schauder’s fixed
point theorem, that % has a fixed point and Theorem 1 is proved for 1,> 0
and a bounded.
(ii) Case of 2 = 0, a Bounded. We use the same argument to let A go
to 0. First let us prove an L” a priori estimate for a solution uj, of
124,+ Q .V.r~i. + a,(ii,)(u,
- ii,,) =f,
ui.l~,wxs”~-
(5’)
=k.
We set Co = 1 + II f II a and we choose C, 2 u, large enough so that the
function v = C, + C,Q . x satisfies, for any I 2 0, M 2 1,
AU
+ Q .VxUj. + aM(6)(v - 6) >A
(9)
vI(iixxs”)- ak.
Then, subtracting (5’) from (9) multiplying
get
IXx9
4”2-V)+
by I iUj,2 L.J, and integrating,
+lxEsN {a~(~j.)(.;-Pi)-a,(i)(V-~)}
we
II fulsc) GO. (10)
The second term of the left-hand side of (10) may be written as
JxxSNCa~(~I)(~~+I:~)-a,(~)(
a,(~,)(u,-u)CQ(.~.,i-Q(
+s
XXS”
+s (l/M+u)Ca,(iij.)-aM(~)lC
XX.9
5X0177/2-14
440
BARDOSETAL.
But on the set {ii,ai7} we have ii,>u,,
since rY~c,>u,,
and thus (A2)
shows that each of these three integrals is nonnegative and, reporting this
in (lo), we obtain that IX, s~(~, - u)+ < 0, and therefore ~4~< u a.e. This
proves:
LEMMA 1. Under the assumptions of Theorem 1 and for M 3 1, the
solution u1 of (5’) satisfies
where K dependsonly on )If I(%, CI,and /Ik )Im.
Remark. In particular, notice that the above estimate is independent
M and holds for any solution of (5) with D . V,u E L’(X x 5”“).
of
To complete the proof of Theorem 1 in the case of 0 bounded, we argue
as in (i). Since 10. is uniformly bounded, 52 .VXuA remains also uniformly
bounded in L”(Xx SN) and thus, by Proposition 1, iin is compact in
L’(X). Extracting a subsequence, u1 converges weakly and II, converges
strongly in L*, and we may pass to the limit in (5’). We do not present the
details since they will be worked out below when M tends to infinity.
(iii) o Unbounded. We denote by u”’ the solution, for a J. > 0, of
hl”+!2Ql”+a,(fi”)(u”-ii”)=f,
~‘+l(<wx~“)- =k.
(5”)
We already know by Lemma 1 that, for any M> 1,
IIUMIIL”,(xxsy G K.
Thus, we have
s
(oM(fiM) u”)” dx dsZ< KPP1
sx
ch4bw1
Pii”dx6C.
(11)
Thus 52 .V,uM is bounded in LP(Xx SN) and by Proposition 1, iiM is compact in L”(X) and also in any Ly(X), 1 Q q < 00, since it is bounded.
With this compactness, we may pass to the limit in every term of (5”).
Indeed, there exists a u E L”(Xx SN) such that, for any q, 1 <q -=zco,
UM
Q .V,U”~
-M
U
weakly in L4(X x SN),
M-rmU
a .v,u
weakly in Ly( X x SN),
strongly in Ly(X).
441
RADIATIVE TRANSFER EQUATIONS
Therefore, for any M’ > 1, inf(M’, o ,Jii”“)) ll(ti, 0j converges strongly in
L*(X) to inf(M’, o(G)) ?I(ti,0) and thus we have
a,(~?~) fan 2 inf(M’,
a,(ii”))
uM II (ir,O) M’“;.
inf(M’, a(C)) u
weakly in L’(Xx
SN).
(12)
By (11) we also know that o,,,(ii”“) uM converge weakly to some function
q(x, Q) E LP(Xx SN) and we have
q(x, Q) 2 inf( M’, a( ii)) u
VM’>l.
Letting M’ go to infinity gives
4(x, Q) 2 o(ii) u.
But clearly
q(x) = lim
o,(ii”)
iiM = o(C) ii,
M-m
and thus
q(x, Q) = o(C) u
Therefore u satisfies
also holds, since the
concludes the proof of
same, using Proposition
a.e.
the first equation of (5). The boundary condition
traces converge weakly in Lp((8Xx SN)) ). This
Theorem 1. The proof of Theorem 2 is exactly the
l(ii) in place of (i), and thus we do not repeat it.
II. ROSSELAND APPROXIMATION
11.1. Setting the Problem
We consider now the resealed model (1) or (2) and are interested in the
limit of U, as E goes to 0. Let us recall that the behavior of U, is known in
various cases. First, in the linear case (a = Cs’), it has been shown that U,
converges strongly to the solution of a linear diffusion equation (see
[3, 7, 12,20,26]). In the nonlinear accretive case (A2) holds for u, = 0)
the same result has been proved [2] (see [Zl] too) and the limit equation
(4) is now a nonlinear diffusion equation. Moreover, since in (4) we have
F(T) = l/[(N+
1) a(T)], this diffusion equation may become degenerate
(F(O) = 0).
For CJ(T) = T-“, (4) is very classical: it is a porous media equation, The
kind of problem has been studied by many authors, and we refer the
442
BARDOSETAL.
interested reader to the papers [4-6, 11, 13, 241 and their references. Let us
point out that the limit u of u,(r, x, Sz) depends only on (t, x). Thus, in
general, a boundary layer appears where the dependence of U, on Q
vanishes exponentially. This problem has been studied, even in the nonlinear case (see [2, 15, 271). Here we will avoid this difficulty by setting
k = c”’ (k is the entering flux) and, for Eq. (1) u0 s u,Jx).
Thus we consider the equations
au,: Q
.vyuc a(&)
-+$ (UC
- C:)=05
at+ E
(13)
or
(14)
where k, is some nonnegative
equations
constant.
We introduce
also the limit
(15)
uIr7x=ko,
u I , = 0 = u,(x),
or
224 - dF(
u) =f,
(16)
ul,,=ko,
where
F(T)=-
1
T ds
-.
N+ 1 s0 a(s)
(17)
Again we state separately our results for the stationary and the evolution
problem.
THEOREM
3 (Rosseland
Approximation
for the Stationary
fa0
Problem).
Under assumption (Al), and (A2) if 2 = 0, let
belong to L”(Xx SN)
and let k, 2 0 be a nonnegative constant. Then any family (u,), ,. of
solutions of (14) in the sense of Theorem 1 is uniformly bounded in
Lco(X x SN). Moreover, we may extract a subsequence(u~,,)“~o which converges pointwise to a function u EL”(X) which is a solution of (16).
RADIATIVE
4 (Rosseland
THEOREM
TRANSFER
443
EQUATIONS
Approximation
for the Evolution
Problem).
Under assumption (Al), let k0 be a nonnegative constant and let u0 Z 0
belong to L”(X).
Then any family (u,),,~ of solutions of (15) in the sense of
Theorem 2 is uniformly bounded in L” (X x SN x R + ). Moreover, we may
extract a subsequence ( uE,),, a 0 which converges pointwise to the unique
solution uEC(R+;
L’(X))nL”(XxR+)
of(15).
11.2. Proof of Theorem 3
Again, we divide our proof in several steps. In the first one, we prove the
uniform L” bounds, then we give some a priori estimates on Q . V,u,. In
the third step, we show that u, is compact in L2(X) and, finally, we pass to
the limit.
(i) L” Bounds for A= 0. First, let us prove an L” estimate for solutions
of (14). Using the same argument as in the proof of Theorem 1, it is
enough to find a supersolution g, with g,a u, (the case L>O is clear
enough). Thus let us introduce the constant y = /I f 11Ix’ (N+ 1) and, since F
is increasing, we may define a function V0 by V, = F- ‘(C, - yx2), where C,
is chosen large enough such that
VO3U,,
VXE
X,
(18)
volax>/ko+
1.
Then, let us define g, for E small enough,
1).
g,;= I’,-&~V,F(V,,)/(N+
We have
and
Q ~Vxx, a,)
-+E2
kc-&)
E
= Q.V,V,
E
D2F( V&2,
4vcJ
Q)+ -Q.V,F(V,,)
E
(N+
1)=2y
This proves, as in Section 1.2, that u, Gg, and we have proved the L”
estimates for A = 0.
(ii) Some a Priori Estimates.
We now prove some a priori estimates
which will be used later to obtain the compactness we need to send E to 0.
444
BARDOS ET AL.
Here, to simplify notations, we set 1 =O. The general case holds with the
same argument.
First we multiply (14) by (~,-k~) and integrate over Xx SN. This gives
1
(u,-k,)‘D~ndo(x)dR+[xXsN~(u~-ii,)2
5 s(rlXXSN)-
and thus
a(&)(
24,
-i7J2
<CE2.
sXx5-N
(19)
Since a(ii,) 2 em, we have the estimates
IIue- k IILZ(X x 9) G C&T
SN)< c.
lI~(li,>O)~-“‘(ii,) 52.V.&, IIL2(XX
(20)
(iii) Strong Convergence of 24,. We use these estimates as follows.
LEMMA 2. Let u, > 0 be bounded in L”(X x SN) independently of E and
Sz.Vxu, E L’(Xx SN). Let B satisfy (Al) and assume (20). Then there exist a
function u E L”(X x SN) and a subsequence u, E u,” such that u, +” _ cuu
pointwise.
Proof:
We deduce from (20) that
=2(o"'(ii,)~,)(a-"~(ii,)52-V,u,9(,~:,,~)
a.e.
(Since &?.V,U,E L’ and uE=O a.e. on the set (GE=O)). But ~r”~(17,) u, is
bounded in L@(Xx SN) (with the same p as in (Al)) since
I
XX.9
(~“~(ii,) u,)*~ dx dQ 6 s
o”(i2,) u,uF-’
XXSN
dx ds2
G lIu,II~-’ s a”(&) ii, dx Q C.
X
Thus (20) shows that Q .V,uf is bounded in LZp’(lfp)(Xx SN). Therefore,
we may apply Proposition 1 and there exists a subsequence uEn (that we
write u,) and a nonnegative function u such that
in any Lp(X), 1 Qp < co.
RADIATIVE
TRANSFER
445
EQUATIONS
Since we have
in L*(X x SN),
2.4;- (i-i,)’ = (24,+ ii,)(un - ii,) -+ 0
we obtain
(%J’~
and, finally,
l<P-r++.
a.e.
u*
this is enough to assert that U, -+ u in any L’(Xx
SN),
(iv) Passage to the Limit. Now, we use this convergence to pass to the
limit in (14). Let us integrate (14) over SN. We obtain
(21)
~~,Q,,l>o,+(lI&)V,.~,Q(~,>Ol=~
Then, multiplying
Combining
(14) by 52 and integrating,
we have
(21) and (21’), we obtain
(22)
In (22), we have the convergences (with the notations
u,- “-CCC u
of (iii))
in L2(X x V),
EjSZTno(fi,)
IU.~O) -0n-cc
in L*(X),
qj-jRGn
-0n-m
iU”~Ol
in L*(X).
To pass to the limit in the last term of (22) we use (20), which shows that
there exists qE L*(Xx SN) such that (extracting again a subsequence)
weakly in L’(Xx
SN).
(23)
446
BARDOS
Multiplying
by u,~~/‘(ii~)
u,V,(QO
Indeed u,u”~(~,)
( u,rJ”2(lQ
ET AL.
we get
Qu,) G
weakly in L’(Xx
q . (d’(u))
converges in L’(Xx
P)
- ufT’~2(u)I < 1u,dqiJ
+O
L’(X x Stg)
P).
to UO”~(U) since
- ii,a~‘2(u”,)( + ( zQP(ii,)
- td2(u)I
(by (Al) and (19)).
Thus, we have proved that
$Vv,(Q@ nu;, - qx(ua”‘(u))
weakly in L’,
in D’( X x P),
-~v,(aoQu)
therefore
q(d2(u))
= p&2
0 mz).
(24)
We now prove that this implies that
(25)
To do so, we work with w = u2 and (24) gives that
II iw,>Ol
1
4
al/2(wli2)=2(N+
1)
G( ~1 V, w,
where
G(z)=
$*q(p)
’
and we know that G is continuous on R+*, G(z) -+;-Ot
C/Z’/~
(by (Al )), and thus we may define
+CO, O<G(z)<
H(t) = j-’ G(s) ds,
0
and we want to show that I (w> o) G(w) V, w = V,H( w) in 9’. But we have
proved that w E Lp(X), V, w E LP(X) for some p > 1 (see (24)), Thus
denoting G,,, = inf(M, G) and H,(t)
= !;I G,,,(s) ds, H, is C’, and we have
V.J,(w)
= II jw>0) G,(w) Vx w. As M tends to infinity, H,(w) converges
pointwise (and thus in any Lq(X)) to H(w). Thus V,H,(w)
converges
RADIATIVE
TRANSFER
441
EQUATIONS
in 9’(X) to V,H(w),
and II in, ,0) G,(w) V, w converges pointwise to
lV.xwl IG(w) Q(w>ojl EL*(X).
a, w>ojG(w)V,w. But I {wzo) IG,(w)V,wl6
Hence, by dominated convergence, the two limits are identical and we have
proved (25).
Therefore, we may pass to the limit in the last term of (22):
-IIn-m
1
1
Q(Ii z-0)
--vv,.Q~~=l{~~>oi-dfi”)
(y2(fi,)
a’,2(iin)
v.r
-QzzL
Q
{u>Ol -=V,F(u)
a”2(u)
(since II IC~,o)/o”2(ii,)
satisfies
-+ Q(U,o)/~1’2(~)
in L*(X)).
u - dF(u)
This
proves that
=Jt
u
(16’)
Finally, u satisfies the boundary condition of (16) since D . V,u% is bounded
in some LP(Xx SN), p > 1. Thus U: has a trace which passes to the limit.
Equation (16’) shows that F(U) also has a trace and they coincide by a density argument. This concludes the proof of Theorem 3.
11.3. Proof of Theorem 4
The main steps of the proof of this theorem are the same as in the
previous one. First the L” bounds are clear since the maximum principle
asserts that
IIu, IIL”(XX
SN) G SUP@,?
IIuoIIL=(xxSNJ.
As before, we obtain the estimate
cT(
ii,)( u, - ii,)2 < a*,
(19’)
i.e.,
II4, - 6, IIL2(XxSNxR+)<CE,
< c.
o-“2(1?,)Q{4,0).
Ii
(20’)
L*(Xx.S"xR+)
We have the
LEMMA
3. With the above notations and assumptions,for any T> 0,
(?)I’* is bounded in L4( [0, T], WB-4(X)) where q = 2p/(p + l), tl= 1 + 1/2p,
foranyB<(p-l1)/(2~+1).
448
BARDOS
Proof of Lemma 3.
L”(Xx
SN). Thus
ET AL.
First proceed as if ~‘/~(ii,)
aua
&~+Q.v.,U,a
Iiat
ui/‘p was bounded in
< c.
IIL*(xx.sNxR+)
Following [ 16, 183, we make a Fourier transform
extension of the functions outside R+ xX). Denoting
variable and ti the Fourier transform of U, we have
in (t, x) (after an
(r, C;) the Fourier
and the last term of the right member of this inequality is bounded
independently of E, 5, r, II so that we may choose, for each (r, 0,
1,
A= 11~12~~~-~/151+E~T/151~~~d~ 1’2
f 16h2 dQ
and we obtain
This would prove that q is bounded in L2( [0, r]; H”‘(X)). But, in fact,
CJ~‘~(U,)u’/*p
c is bounded in L2P( [0, r] x Xx SN), so that one only knows
that
RADIATIVE
TRANSFER
449
EQUATIONS
is bounded in L2p’(Pf *)( [0, T] x Xx SN). Consider now the map Y:
where
f -)q
From
the above computation,
we know that Y maps La(.s>O;
r] xXx SN)) into LCO(E>O; L*([O, T]; H”‘(X)));
it also maps
obviously L”(E > 0; L’( [0, T] x Xx SN)) into LOO(.s> 0; L’( [0, T] x X)).
Therefore by a standard interpolation
argument, we know that z is
bounded in Lzp’(p+‘)([O, T]; W s,2pm+ ‘j(X)) for any s < (p - 1)/2p. It
remains to show that if u E Vy( RN + ’ ) and u has a compact support, then
&I E WbT([WNf 1), for 0 < fl <s/u. We may compute
L*([O,
1ul’yx + h) - P(x)y
lhl N+l+yS
where B= {hERN+’
dxdh+
/I h) G 1 }. But we have, since l/cr < 1,
IIu”aII“w.,< c 5
RN+‘,
B
I u(x + h) - o(x)(~”
N+l+y/l
lhl
dxdh+C()uJy,,)“”
1u(x + h) - 11(x)1” “’
<C
lhl N+l+sy
6 c IIuIIg& + c 11
0 11
(ILlyr
.
U
This
proves
that
(tP(yLr,
RNtlxB
(2)“”
is bounded
+ cwy,P
in Ly([O, T];
WP.y(X))
with
q = 2p/( p + 1) and /I -C(p - 1)/2p(r, and Lemma 3 is proved.
Remark.
To write the Fourier transform we have used several times
extensions of U, to the whole space. This is possible thanks to the results of
[9] and the assumption on uEICdXrS~J_, u,\,=~. We may assume that these
extensions have the same compact support.
Let us continue the proof of Theorem 4. Following
the previous section, we write
a
at 4 - v,
Thus (20’) shows that X,/at
ax
E$+V,SiE
the computation
>I=0.
is bounded in Ly([O, T]; WPIJJ(X)).
of
(26)
450
BARDOS ET AL,
Setting 0, = (q)“‘,
we have proved that
u,;E LY( [O, T-J; wyx)),
I~~:--V,:lLu~~“,T]rX)
Tz+O
$
(since 2?,, v,: are bounded)
(27)
LY([O,T]; w- ‘,“(X))
with uniform bounds in E. The next step of our proof is to deduce from (27)
that ii, has a subsequence which converges in Lq( [0, T] x X).
From (27), we deduce that z?, has a subsequence which converges in
Ly( [0, ZJ x A’), according to the following lemma.
LEMMA.
Let V c WC Z he three Banach spaces, and assume that the
inclusion V c W is compact, and the inclusion WC Z continuous. Let fx and
g, he families such that
- f, is bounded in Ly([O, T]; V);
- fE-gC+O in Ly([O, T]; W) when ~-0;
~- a, g, is bounded in LY( [O, T]; Z).
Then fz is compact in Ly( [0, T]; W).
Indeed, we apply this lemma to V= IV”(X),
W= L”(X),
and
z= w-‘,Y(X).
This lemma is almost classical, but we give the proof of it for the sake of
completeness.
Proof of Lemma. We can obviously assume that we are in the situation
where V c Y c WC Z, where the inclusions V c Y and Y c W are compact.
Define
$(t,
h) =fXt
Since the inclusion
that
+ h) -A;(t);
eJt, h) = g,(t + h) - g,(t).
V c Y is compact, for any c1> 0, there exists C, > 0 such
Replacing o; by f;+ h a,g,(s) ds, one has, by integration
inequality on any compact t-set,
of the above
RADIATIVE TRANSFER EQUATIONS
451
6 Colp Iu;(t, h)ly, dt
s
+ cc;
< col+ cq
lw;(t,h)-~,(z,h)l&dt+h4~‘~~r+hl~,gi(~)lPd~dt~
I
/9(E) + C/6).
The latter estimate ensures the compactness of f, in Ly( [0, T]; IV), by
using Kolmogorov’s
LY-compactness criterion derived from the Ascoli
theorem, and the compactness of the inclusion YC W.
The end of the proof of Theorem 4 is now the same as point (iv) of the
proof of Theorem 3 and we do not repeat it.
III.
EXISTENCE OF SOLUTIONS FOR THE MODEL
WITH FREQUENCIES
In this section, we consider the complete model of Radiative Transfer.
Thus we consider the solution T(x), u(x, Q, v), x E X, Q E S”‘, and v E [w+,
of the system
AT+
sIw+~v(TM&(T)-4dv=g,
We extend the results of Section I to (28) and prove the existence of
solutions even for a singular opacity (a,(O) = + co). Again we begin by
studying a “regular case” where 0” is regularized (Section III.1 ), then we
treat the general case (Section 111.2).
To simplify the proofs we only consider the case I > 0.
111.1. Regular Opacities
Let us introduce some assumptions on the opacities (a,) and the
Planckian reemission (B,,). We will assume that there exists a positive
number 6 and two functions C,, P, for v > 0 such that
PJ,E
L’(R+),
(29)
O~B,(T)~P,,O~~,(T)~C,,VTE[O,~]VVZO,
B,( . ), a,( .) are continuous
for a.e. v Z 0,
(30)
(31)
452
BARDOS
ET AL.
0<f(x, a, v), k(x, Q, v)6 1P,#,0 <g(x) < 16,
(32)
for a.e. v, the functions f, k, a,(T), a,,(T) B,( 7’) are continuous
with respect to frequencies at point v uniformly for TE [0, S],
XEX, s2ESN.
(33)
Finally, beyond these technical assumptions, we need a fundamental
assumption to be able to solve the second equation of the system (28). The
most general one seems to be
for each measurable function w(v), 0 6 w(v) < P,, there exists a
unique measurable function D,,(x) E [0, S] s.t.
I@,. + jom a,(@,)
B,(@,)
dv = Ia CJ~(@,+,)w dv +g.
0
Remarks.
(1) Assumption (34) is satisfied in particular when the
function a,(@) B,(Q) is nondecreasing in @ and ~“(a) nonincreasing in @
for a.e. v. This case corresponds to accretive operators; see [22, 171. We
can even generalize this by assuming only that j,” G,(Q) B,(G) dv is nondecreasing.
(2) The assumptions (29)-(33) are compatible with Planck’s values
of B, and Kramer’s values of 0” with uniformly positive temperature. The
case of zero temperature is treated in Section 111.2. Let us recall that Planck
and Kramer’s functions are (assuming that the local temperature and
energy are proportional)
(B)
(Cl
and the assumption
(34) holds since Remark 1 applies.
(3) Here we have assumed that v E IX+ since it is the physically
relevant case. For numerical applications, it is useful to assume that v
belongs to some discrete set. This subsection holds in this case with very
few changes.
Let us state the main result of this subsection.
PROPOSITION
2. Under assumptions
solution (u, T) of (28) and it satisfies
(29 j(34)
there exists at least one
O<udP,,OdT<&
and (28) holds in the sense that Q .V,UE Lm(X x SN) for a.e. v.
453
RADIATIVE TRANSFER EQUATIONS
Proof. The proof of Proposition 2 uses the same arguments as the proof
of Theorem 2 and thus we only prove the new points. We introduce the set
w= {@EL’(X),O<@<6)
(with its L’ topology).
For any @E%, we may solve the linear transport equation with parameter
V:
Au + Q .v,u + a,(@) u = a,(@) B,(Q) +f,
(35)
Ul,dXxSN,m-- k.
The maximum
principle gives
O<udP,,
since we have with a simple calculation
(36)
that
A(u-P,)+Q.V,(u-P,)++o,(@)(u-P,)+
~fl,,(@HW@)-Pv)
Q(u>P,,}+4f-P,)
Q(u3P,.)
GO
and
(u-p”)+
I (cwxSs”)_60.
Following Section I, we set ii = T, @, and, using (34), we may define an
operator T2 which associates to w(x, v), 0 d w(x, v) < P,, the solution Y(x),
o< Y(x)<&
of
To solve (28) is now equivalent to finding a fixed point for T= T, . T, and
thus it is enough to prove that T is compact from $9 to 9.
To this end, we introduce the set
d={wEL&(XxRf),
with the topology of Lk(Xx
O<wdB,},
lR+), i.e., L’ with weight C,
IbIL:=jx,,+Iw(x, v)l C, dx dv,
and our theorem may be reduced to the following lemmas.
LEMMA
4.
T, is continuous compact from % into 9.
454
LEMMA
BARDOS
5.
ET AL
T, is continuous from 9 into W.
Proof of Lemma 4. The continuity of T, is deduced from the continuity
of the solution of (35) with respect to @ for every v and from the
dominated convergence (using (29) and (30)); we leave it to the reader and
we prove the compactness. We divide this proof in three steps.
First Step.
Continuity
of T, CJ in v
We will need this continuity,
and more precisely that
sup II T, @(A v) - T, @(x v’)ll LOX) 1’.
0
otx
for a.e. v.
(37)
To prove (37), take v such that (33) holds and set w(x, a) = u(x, Q, v) u(x, Q, v’) for a solution u of (35). Then
i.e.,
lu’l +Q.V, IU’I <sup IL-fdl
‘, R
+ SUP
@Cb
After integration,
+;t$
Io,B,,(~)-a,,B,,.(~)l
Ia,,B,(@D)
- a,,,&(@)1 + sup I a,,(@) - a,(@)[ P,,,.
@56
we obtain
5x x s” I u’l dx dQ d C sup-I.D {If,
-f,., 1+ Ik,. - k,., I
Io,B,(@) - a,,,B,,,(@)l
cfJG6
+ SUP
+ sup I a,,(@)-a,.(@)[
a<6
By (33), the second member of this inequality
proved.
Second Step.
P,,,}.
converges to 0 and (37) is
Compactness of T, @ for a.e. v.
Now, we prove that for a.e. v, the family (T, @)aE w is relatively compact.
For a solution u of (35), we have
Q.V,u= -Au-o,(@)u+f+a,B,(@).
RADIATIVE
Introducing
TRANSFER
the solution u E L”(Xx
455
EQUATIONS
SN) of (v is a parameter of u)
sz~v,u=o,
uI(dXXSN)-= k
we obtain
(u - u) I (dXX SN)- = 0.
Hence, for a.e. v, 52 . V,(u - u) remains bounded in Lm(X x 9”) and, by
Proposition 1, it shows that the family (U’(X))@~~ is relatively compact in
L’(X) (and thus in L’(X)).
Third Step.
Compactness of T, in 9.
Now, let us choose a dense family (v~),,~ wIfor which (33) holds. For any
sequence 0” E %?and W” = T, W, we may extract by a diagonal procedure
(and using step 2) a subsequence, still denoted wn, such that for some
4% v,),
WY& VA) n--+ool w(x, Vh)
in L’(X)
Vv,.
Using the first step, we deduce that w~(x, v) converges in L’(X) to some
w(x, v) for a.e. v, and thus, by dominated convergence, we obtain that
5Iw+ z, Ix lw”-ww(
dxdv-0. n-cc
This proves the compactness of T, and it remains to prove Lemma 5.
Proof of Lemma 5. Let a sequence w” E 9 converge to w in L-k, then it
converges a.e. (extracting subsequence). By compactness, for a.e. x E A’, we
may extract from Yn = T2(w”) a subsequence Y’(x) which converges to
some Y(x) E [0, S]. Passing to the limit in (34), we obtain that Y(x) is the
unique solution of
WI
+ Ioa a,( ‘J”(x)) B,( Wx)) dv = joa 0°C Y(x)) w(v, -y) dv + g(x).
This shows that for a.e. x, the full sequence Y’(x) converges to Y(x) = T, w
in L’(X)
and the proof of Lemma 2 is complete, thus proving
Proposition 2.
111.2. Singular Opacities
In this section, we treat the case of a more general opacity than in
Section 111.1. Mainly we extend Proposition 2 to the case where g,(T)
580/77/2-IS
456
BARDOS ETAL.
blows up as T+ 0 (a,(O) = + co) in order to admit the Planck’s function
B, and Kramer’s opacity 0” given in Remark (2).
We could give a set of general assumptions for which the following
theorem holds, but these assumptions are rather technical and numerous.
Thus we prefer to state our results for the particular case of Remark (2).
First, we introduce a “regularized” version of (28) by setting
%u,+s;z.V.,u,+a,(T,)u,=a,,(T,)B,(T,)+,f;
Since, in the case of Remark (2), s a,(T) B,(T) dv is increasing and a,(T) is
decreasing, we obtain an a priori lower bound on T,, T, b T,,,(M), where
T,,,;,(a) is the solution of
JTmin(@) + joa’ CT,.B,( T,,,,,(a)) dv = H.
This allows us to apply Proposition
(u,, T,) which satisfies
2 and thus (28), has a solution
0 d u, d B,(6 + LY)= P,,,
O<T,<6+cr,
(38)
whenf, g, k satisfy (32).
Then, our goal is to prove the
THEOREM 5. We assume that B,, CS~are given by (B)-(C), and thatf, g,
k satisfy (32), and that f and k are continuous in v, for a.e. v, uniformly for
XEX, QESN. Denote (Us, T,) the solution of (28), obtained through
Proposition 2. Then we may extract from (u,, T,) a subsequence (u,, T,,) s.t.
u,-u
in L”(Xx SN x R+) weak-* and T, -+ T in L’(X) and (u, T) is a
solution of (28). (u, T) satisfy (38) and (28) holds in the sense that
Cl.V,u~L”(XxS~)for
a.e. v and o,(T) ~EL~(XXS~;
L’(R:))for
some
q> 1.
Remark.
Again we use the convention
a,(O) .O = 0.
Proof: As before, we first prove some compactness and we pass to the
limit in (28), using this compactness.
First Step.
Compactness of sy tl, dv.
Denoting w,(x, Q) = s u, dv, our goal is to prove that 9, is compact in
L’(X). w, satisfies the equation
sz~v,w,+jR+ M2u,dv=jR+
(~,&(Ta)+f-k)dv,
(39)
RADIATIVE
and, by Proposition
that
TRANSFER
451
EQUATIONS
1, it is enough to prove that there exists q > 1 such
CJ .V.,W,E LY(Xx
SN).
(40)
The right side of (39) is clearly in this set and we compute, for any q > 1,
s> 1,
1 (I
xxs”
R’
d
a,(T,)u,dv
sXx.‘?
>
qd.xdQ
df2 dx
<C
o;(TJ
u,dv (j-+
k-l
dv)‘-.‘1
(T;( T,) ii, dv dx
6C av(T,)
6 xx~v(Td(I)/($
-- 1) s.(a)ll’
l)‘$
One can check using assumptions (B)-(C) that this remains bounded by
choosing q, s > 1 properly. In the following we call q such a choice.
Then, we have obtained that G2 is compact in L’(X) and in any Lp(X),
l<p<+co.
Second Step. Compactness of ii,(x, v).
We now prove that u’, is compact in L’(X x [w + ).
Using step 1 and (38) (with P,E L’(R+))
and step 1 of the proof of
Lemma 4, it remains to estimate
I s
XxSN
1u,(x, Q, v + h) - u,(x, 52, v)l dv dx dQ
v,h
<C
sup Clf”-f”+kl+
I&-k+,Il
.x. R
+sTup6 IOVB,(T)-a,+,B,+,(T)I
+ f J-“,,I I x ‘O v+h(TJ-u,(T,)I
I
dv
Ux,v)dvdx.
458
BARDOS
ET AL.
Thus there exists a modulus of continuity
this inequality is bounded by
where
P”(E)
+E400,
k
~‘(5
h)
p such that the right-hand
side of
0 (use the special form (B)-(C)).
+h+O
Thus we have obtained
I i&(x,
IS
x I’E R+
<2
v + h) - z&(x, v)l dv dx
P,, dv +p(h)
+ in& {P”(E) + P’(E, h)} m
and have proved the compactness of t?, in L’(Xx
0
R + ).
Third Step. Compactness of TM.
Therefore, we may extract from ii, a subsequence denoted ii, which converges a.e. to u(x, v). Since we have
AT, + I,+ u,&(T,J dv = j
a,(T,J Ux, v) dv+g(x),
lQ+
we check, as in the proof of Lemma 5, that T,,(x) converges to T(x) (for
a.e. X) the unique solution of
l.T+jR+a,B,(T)dv=j
Iw*
a,(T)o(x,v)dv+g(x).
(41)
Fourth Step. Passage to the limit.
Following Section I, we may extract from U, a sequence (still denoted u,)
such that
459
RADIATIVE TRANSFER EQUATIONS
a, Q,VI+ 4x,Q2,
v)
in L”(Xx
i&(x, v) * qx, v) = u(x, v)
Q .v*u,
aA
in L’(Xx
SN x IX+) weak-*,
R’),
+ Q .v,u
in&V(XxS”xR+),
-+ U.BY(T)
inL’(Xx
R+).
Moreover,
we know that (41) holds and it remains to prove that o,( r,,) u,,
converges to u,(T) U.
Since a,( r,) u, is nonnegative and bounded in L’(X x SN x R + ), it converges weakly to a nonnegative measure m on Xx SN x R+. On the other
hand, for any M> 1, inf(M, a,(T,,)) II (T,Oj converges, as n tends to
infinity, to inf(M, a,(T)) I jT,Oj for a.e. x, v. Thus, for any M’ > 1, we have
o,(T,)
42
WM
o,(TJ)
4
Q frroj
3
inf(M, a,(T)) u
weaklyinLP(XxSNx[O,MI]),l<P<~.
Therefore, we have obtained (sending A4 to infinity)
m>,cr,(T)ul.,o=a,(T)u,
but, since
we have m = G,( 7’) u and Theorem 5 is proved.
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